Sizes of witnesses in Covtree

TL;DR

This paper investigates the size bounds of minimal witnesses in poset structures related to quantum gravity, introducing the exchange graph as a new analytical tool and establishing specific size bounds.

Contribution

It introduces the exchange graph of downsets and proves new bounds on the size of minimal witnesses, including a non-linear upper bound and specific cases for k=3.

Findings

No linear upper bound of the form n+k+c exists for minimal witnesses.

All minimal witnesses satisfy |Q| ≤ nk - n.

For k=3, there exists a minimal witness with size ≤ 1.5(n+1).

Abstract

Given a set $\Gamma$ of $k$ unlabelled posets, each of size $n$, we say that a poset $Q$ is a \emph{witness} to $\Gamma$ if $\Gamma$ is the set of downsets of size $n$ of $Q$. We say that $Q$ is a \emph{minimal witness} if it does not contain a proper downset that is itself a witness to $\Gamma$. Motivated by the causal set approach to quantum gravity, we study the upper bound on the size of minimal witnesses as a function of $n$ and $k$. We show that there is no linear upper bound of the form $n+k+c$ for any constant $c$. We introduce the \emph{exchange graph of downsets} as a new tool to study this scenario, and use it to show that all minimal witnesses $Q$ satisfy the bound $|Q|\leq nk-n$, and that when $k=3$ there is at least one minimal witness $Q$ that satisfies the bound $|Q|\leq \frac{3}{2}(n+1)$.